Noncommutative geometry (NCG) is a branch ofmathematics concerned with a geometric approach tononcommutative algebras, and with the construction ofspaces that are locally presented by noncommutative algebras of functions, possibly in some generalized sense. A noncommutative algebra is anassociative algebra in which the multiplication is notcommutative, that is, for which${\displaystyle xy}$ does not always equal${\displaystyle yx}$; or more generally analgebraic structure in which one of the principalbinary operations is not commutative; one also allows additional structures, e.g.topology ornorm, to be possibly carried by the noncommutative algebra of functions.

An approach giving deep insight about noncommutative spaces is throughoperator algebras, that is, algebras ofbounded linear operators on aHilbert space.^[1] Perhaps one of the typical examples of a noncommutative space is the "noncommutative torus", which played a key role in the early development of this field in the 1980s and lead to noncommutative versions ofvector bundles,connections,curvature, etc.^[2]

The main motivation is to extend the commutative duality between spaces and functions to the noncommutative setting. In mathematics,spaces, which are geometric in nature, can be related to numericalfunctions on them. In general, such functions will form acommutative ring. For instance, one may take the ringC(X) ofcontinuouscomplex-valued functions on atopological spaceX. In many cases (e.g., ifX is acompactHausdorff space), we can recoverX fromC(X), and therefore it makes some sense to say thatX hascommutative topology.

More specifically, in topology, compactHausdorff topological spaces can be reconstructed from theBanach algebra of functions on the space (Gelfand–Naimark). In commutativealgebraic geometry,algebraic schemes are locally prime spectra of commutative unital rings (A. Grothendieck), and every quasi-separated scheme${\displaystyle X}$ can be reconstructed up to isomorphism of schemes from the category of quasicoherent sheaves of${\displaystyle O_X}$-modules (P. Gabriel–A. Rosenberg). ForGrothendieck topologies, the cohomological properties of a site are invariants of the corresponding category of sheaves of sets viewed abstractly as atopos (A. Grothendieck). In all these cases, a space is reconstructed from the algebra of functions or its categorified version—somecategory of sheaves on that space.

Functions on a topological space can be multiplied and added pointwise hence they form a commutative algebra; in fact these operations are local in the topology of the base space, hence the functions form a sheaf of commutative rings over the base space.

The dream of noncommutative geometry is to generalize this duality to the duality between noncommutative algebras, or sheaves of noncommutative algebras, or sheaf-like noncommutative algebraic or operator-algebraic structures, and geometric entities of certain kinds, and give an interaction between the algebraic and geometric description of those via this duality.

Regarding that the commutative rings correspond to usual affine schemes, and commutativeC*-algebras to usual topological spaces, the extension to noncommutative rings and algebras requires non-trivial generalization oftopological spaces as "non-commutative spaces". For this reason there is some talk aboutnon-commutative topology, though the term also has other meanings.

There is an influence of physics on noncommutative geometry.^[3] Thefuzzy sphere has been used to study the emergence ofconformal symmetry in the 3-dimensionalIsing model.^[4]

Some of the theory developed byAlain Connes to handle noncommutative geometry at a technical level has roots in older attempts, in particular inergodic theory. The proposal ofGeorge Mackey to create avirtual subgroup theory, with respect to which ergodicgroup actions would becomehomogeneous spaces of an extended kind, has by now been subsumed.

The (formal) duals ofnon-commutativeC*-algebras are often now called non-commutative spaces. This is by analogy with theGelfand representation, which shows thatcommutative C*-algebras aredual tolocally compactHausdorff spaces. In general, one can associate to any C*-algebraS a topological spaceŜ; seespectrum of a C*-algebra.

For theduality between localizablemeasure spaces and commutativevon Neumann algebras,noncommutativevon Neumann algebras are callednon-commutativemeasure spaces.

A smoothRiemannian manifoldM is atopological space with a lot of extra structure. From its algebra of continuous functionsC(M), we only recoverM topologically. The algebraic invariant that recovers the Riemannian structure is aspectral triple. It is constructed from a smooth vector bundleE overM, e.g. the exterior algebra bundle. The Hilbert spaceL²(M, E) of square integrable sections ofE carries a representation ofC(M) by multiplication operators, and we consider an unbounded operatorD inL²(M, E) with compact resolvent (e.g. thesignature operator), such that the commutators [D, f] are bounded wheneverf is smooth. A deep theorem^[5] states thatM as a Riemannian manifold can be recovered from this data.

This suggests that one might define a noncommutative Riemannian manifold as aspectral triple (A, H, D), consisting of a representation of a C*-algebraA on a Hilbert spaceH, together with an unbounded operatorD onH, with compact resolvent, such that [D, a] is bounded for alla in some dense subalgebra ofA. Research in spectral triples is very active, and many examples of noncommutative manifolds have been constructed.

In analogy to theduality betweenaffine schemes andcommutative rings, we define a category ofnoncommutative affine schemes as the dual of the category of associative unital rings. There are certain analogues of Zariski topology in that context so that one can glue such affine schemes to more general objects.

There are also generalizations of the Cone and of the Proj of a commutative graded ring, mimicking a theorem ofSerre on Proj. Namely the category of quasicoherent sheaves of O-modules on a Proj of a commutative graded algebra is equivalent to the category of graded modules over the ring localized on Serre's subcategory of graded modules of finite length; there is also analogous theorem for coherent sheaves when the algebra is Noetherian. This theorem is extended as a definition ofnoncommutative projective geometry byMichael Artin and J. J. Zhang,^[6] who add also some general ring-theoretic conditions (e.g. Artin–Schelter regularity).

Many properties of projective schemes extend to this context. For example, there exists an analog of the celebratedSerre duality for noncommutative projective schemes of Artin and Zhang.^[7]

A. L. Rosenberg has created a rather general relative concept ofnoncommutative quasicompact scheme (over a base category), abstracting Grothendieck's study of morphisms of schemes and covers in terms of categories of quasicoherent sheaves and flat localization functors.^[8] There is also another interesting approach via localization theory, due toFred Van Oystaeyen, Luc Willaert and Alain Verschoren, where the main concept is that of aschematic algebra.^[9][10]

Some of the motivating questions of the theory are concerned with extending knowntopological invariants to formal duals of noncommutative (operator) algebras and other replacements and candidates for noncommutative spaces. One of the main starting points ofAlain Connes' direction in noncommutative geometry is his discovery of a new homology theory associated to noncommutative associative algebras and noncommutative operator algebras, namely thecyclic homology and its relations to thealgebraic K-theory (primarily via Connes–Chern character map).

The theory ofcharacteristic classes of smooth manifolds has been extended to spectral triples, employing the tools of operatorK-theory andcyclic cohomology. Several generalizations of now-classicalindex theorems allow for effective extraction of numerical invariants from spectral triples. The fundamental characteristic class in cyclic cohomology, theJLO cocycle, generalizes the classicalChern character.

• In thephase space formulation of quantum mechanics, thesymplecticphase space ofclassical mechanics isdeformed into a non-commutative phase space generated by theposition and momentum operators.
• Thenoncommutative torus, deformation of the function algebra of the ordinary torus, can be given the structure of a spectral triple. This class of examples has been studied intensively and still functions as a test case for more complicated situations.
• Snyder space^[11]
• Noncommutative algebras arising fromfoliations.
• Examples related todynamical systems arising fromnumber theory, such as theGauss shift on continued fractions, give rise to noncommutative algebras that appear to have interesting noncommutative geometries.

This sectionneeds expansion. You can help byadding missing information.(May 2023)

This sectionneeds expansion. You can help byadding missing information.(May 2023)

AConnes connection is a noncommutative generalization of aconnection indifferential geometry. It was introduced byAlain Connes, and was later generalized byJoachim Cuntz andDaniel Quillen.

Given a rightA-moduleE, a Connes connection onE is a linear map

${\displaystyle \mathrm{\nabla }:E\to E{\otimes }_A{\mathrm{\Omega }}^{1}A}$

that satisfies theLeibniz rule${\displaystyle {\mathrm{\nabla }}_r(sa)={\mathrm{\nabla }}_r(s)a+s\otimes da}$.^[12]

• Commutativity
• Fuzzy sphere
• Koszul connection
• Moyal product
• Noncommutative algebraic geometry
• Noncommutative topology
• Phase space formulation
• Quasi-free algebra

• Consani, Caterina;Connes, Alain, eds. (2011),Noncommutative geometry, arithmetic, and related topics. Proceedings of the 21st meeting of the Japan-U.S. Mathematics Institute (JAMI) held at Johns Hopkins University, Baltimore, MD, USA, March 23–26, 2009, Baltimore, MD: Johns Hopkins University Press,ISBN 978-1-4214-0352-6,Zbl 1245.00040
• van Suijlekom, Walter (2025).Noncommutative Geometry and Particle Physics. Mathematical Physics Studies.Springer Nature.doi:10.1007/978-3-031-59120-4.ISBN 978-3-031-59120-4.

Wikiquote has quotations related toNoncommutative geometry.

• Introduction to Quantum Geometry by Micho Đurđevich
• Ginzburg, Victor (2005). "Lectures on Noncommutative Geometry".arXiv:math/0506603.
• Khalkhali, Masoud (2004). "Very Basic Noncommutative Geometry".arXiv:math/0408416.
• Marcolli, Matilde (2004). "Lectures on Arithmetic Noncommutative Geometry".arXiv:math/0409520.
• Madore, J. (2000). "Noncommutative Geometry for Pedestrians".Classical and Quantum Nonlocality: 111.arXiv:gr-qc/9906059.Bibcode:2000cqnl.conf..111M.doi:10.1142/9789812792938_0007.ISBN 978-981-02-4296-1.S2CID 15595586.
• Masson, Thierry (2006). "An informal introduction to the ideas and concepts of noncommutative geometry".arXiv:math-ph/0612012. (An easier introduction that is still rather technical)
• Noncommutative geometry on arxiv.org
• MathOverflow,Theories of Noncommutative Geometry
• Mahanta, Snigdhayan (2005). "On some approaches towards non-commutative algebraic geometry".arXiv:math/0501166.
• Sardanashvily, G. (2009). "Lectures on Differential Geometry of Modules and Rings".arXiv:0910.1515 [math-ph].
• Noncommutative geometry and particle physics
• connection in noncommutative geometry in nLab

• Connection (mathematics)
• Differential geometry
• Noncommutative geometry
• Mathematical quantization
• Quantum gravity

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